Unique polynomial through n+1 points. Lagrange form, Newton divided differences. Error bound. Least squares approximation.
graph TD
D1["Def: Lagrange basis\nLⱼ(xᵢ)=δᵢⱼ"]
D2["Def: Divided difference\nf[x₀,...,xₙ]"]
D3["Def: Best approx\nmin ‖f−p‖"]
T1["Thm: Uniqueness\ndeg ≤ n through n+1 pts"]
T2["Thm: Lagrange form\np = Σ fⱼ Lⱼ"]
T3["Thm: Newton form\np = f[x₀]+...+f[x₀..xₙ]πⱼ"]
T4["Thm: Error bound\nf−p = f⁽ⁿ⁺¹⁾(ξ)π/ⁿ⁺¹!"]
T5["Thm: Least squares\nmin Σ(f−p)² ⇒ normal eqs"]
L1["Lemma: Runge phenomenon"]
D1 --> T2
D2 --> T3
T1 --> T2
T2 --> T4
T3 --> T4
D3 --> T5
T4 --> L1
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
classDef lemma fill:#74c0fc,color:#fff
class D1,D2,D3 definition
class T1,T2,T3,T4,T5 theorem
class L1 lemma
Process Statistics
- Nodes: 15
- Edges: 12
- Definitions: 3
- Theorems: 5
- Lemmas: 1
Frontier: Radial basis functions, splines, machine learning surrogate models. math.NA